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Extending characterizations of truthful mechanisms from subdomains to domains. Angelina Vidali angelina.vidali@univie.ac.at Theory and Applications of Algorithms Research Group University of Vienna, Austria Abstract. The already extended literature in combinatorial auctions, public projects and scheduling demands a more systematic classiﬁcation of the domains and a clear comparison of the results known. Connecting characterization results for diﬀerent settings and providing a characterization proof using another characterization result as a black box without having to repeat a tediously similar proof is not only elegant and desirable, but also greatly enhances our intuition and provides a classiﬁcation of diﬀerent results and a uniﬁed and deeper understanding. Characterizing the mechanisms for the domains of combinatorial auctions and scheduling unrelated machines are two outstanding problems in mechanism design. Since the scheduling domain is essentially the subdomain of combinatorial auc- tions with additive valuations, we consider whether one can extend a characterization of a subdomain to a domain. This is possible for two players (and for n-player mechanisms that satisfy stabilty) if the only truthful mechanisms for the sub-domain are the aﬃne maximizers. Although this is not true for scheduling because besides the aﬃne maximizers there are other truthful mechanisms (the threshold mechanisms), we still show that the truthful mechanisms that allocate all goods of practi- cally any domain which is strictly superdomain of additive combinatorial auctions are only the aﬃne maximizers. 1 Introduction Our results and motivation. Roberts [12](1979) gives an elegant proof, which shows that the only truthful mechanisms for the Unrestricted domain are the aﬃne maximizers. He also gets the Gibbard-Sattherwhaite Theorem (1973) for voting systems as a corollary. For more “re- stricted” multi-parameter domains, there exist truthful mechanisms other than aﬃne maximizers (see e.g. [16, 11, 4]). An important question, posed in [19, 15], is to determine how much we need to restrict the domain in order to admit truthful mechanisms diﬀerent than the aﬃne maximizers. Here we show that for the case of two players, the transition domain is the additive combina- torial auctions(/scheduling) domain: We show that if we slightly enrich the possible valuations, the threshold mechanisms involved in the characterization [4] seize to be truthful and the only truthful mechanisms that remain are the aﬃne maximizers. In this work we address but only partially answer the following very important strengthenings of this question: In which way should we restrict the domain? Which domains have the same characterization? Can we classify the domains in a hierarchy in terms of how diﬃcult it is to characterize them (if their characterization is the same) and how rich are the mechanisms allowed (else)? Every time we achieve to characterize a more diﬃcult domain do we automatically get a proof for domains that are lower in this hierarchy? For which domains can we establish a bijection between the mechanisms involved in their characterization? This paper gives some explanations we would have liked to ﬁnd, back when we started working on characterization results and wondered what do the results about other slightly diﬀerent domains tell us about the domain we were primarily interested in. A crucial observation is: the more “unrestricted” the domain of valuations, the fewer the pos- sible truthful mechanisms. An intuitive explanation for this is that in larger domains there are more inputs that need to satisfy the conditions for truthfulness. On the other hand, this intuition may be misleading: Given that a sub-domain admits as truthful mechanisms only the aﬃne maxi- mizers does not immediately imply that the domain also admits the same mechanisms; there may be other mechanisms which when restricted to sub-domain are exactly the aﬃne maximizers. In particular, we don’t know whether this is possible for more that 2 players. However, for the case of 2 payers we verify this intuition: A complete characterization for the scheduling problem, where the valuations are heavily restricted to additive ones, involves a combination of aﬃne minimiz- ers and threshold mechanisms. On the other hand the characterization for all it’s super-domains can be easily derived from it; this derivation is much easier and clearer and involves only aﬃne maximizers. We provide a single characterization proof for any super-domain of this slight enrichment of the additive domain. One of these super-domains is the domain of 2-player combinatorial auctions with sub-modular valuations (that allocate all items), an important domain about which no characterization was previously known, but also the already known characterizations for 2- player combinatorial auctions [16] and combinatorial auctions with sub-additive valuations [11]. Our approach also goes through for n-player stable mechanisms. Our work proposes a common general framework that classiﬁes a multitude of diﬀerent do- mains. If you prove a characterization of truthful mechanisms for a speciﬁc (2-player that allocates all items, or n-player stable) domain in terms of aﬃne maximizers and threshold mechanisms, you can plug your Theorem as black box in our theorems here and get a characterization of all its super-domains. So it is more important to characterize a domain with a rich class of super- domains. The notions of translated domains and bijections between domains and the tools we develop might be further useful. Related work. The starting point of characterization attempts goes back to Robert’s [12] re- sult. Many papers tried to extend this very elegant proof [17, 11, 20], while others tried diﬀerent proof techniques [16, 4, 9, 17]. (As the literature in combinatorial auctions is vast we refer the reader to [19] Chapter 11 and the references within and mention here only some recent results.) An important direction is the quest for polynomial-time algorithms. Computational complexity impossibility results for maximal in range mechanisms where shown in [2, 8]. Dobzinski [7] shows that every universally truthful randomized mechanism for combinatorial auctions with submodu- 1 lar valuations that provides an approximation ratio of m 2 − must use exponentially many value queries. Krysta and Ventre show that if veriﬁcation is introduced sub-modular combinatorial auc- tions become tractable [14]. Many more interesting results arise when one considers randomized mechanisms. Another very well-studied relevant problem is that of multi-unit auctions, and one of our proofs here goes along the same lines as a proof from [6]. Nisan and Ronen introduced the mechanism-design version of the scheduling problem on unrelated machines [18, 5, 13]. For the case of two machines [11] Dobzinski and Sundararajan characterized all mechanisms with ﬁnite approximation ratio for the objective of minimizing the makespan, while [4] gave a characterization regardless of approximation ratio of decisive truthful mechanisms (which also implies a characterization of additive combinatorial auctions that we will use here) in terms of aﬃne minimizers and threshold mechanisms. 1.1 Deﬁnitions and preliminaries Stability is without loss of generality for 2-player auctions that allocate all items, unrestricted domains and combinatorial public projects. A mechanism is called stable if the following holds: For ﬁxed valuations v−i , the allocation ai of player i determines uniquely the allocation a−i of the other players. (In other words: Fix v−i , then for all vi for which player i has allocation ai the allocation a−i is the same.) Stability can be assumed without loss of generality for unrestricted domains, combinatorial public projects and 2-player auctions where all items are allocated. It is too restrictive for combinatorial auctions with n ≥ 3 players (see [16] Example 4), however all known characterization results [12, 16, 20, 11, 4, 17, 11] heavily rely on stability, or characterize domains where stability can be assumed essentially without loss of generality. Stability is implied by S-MON or IIA (see [16, 1, 11] for a discussion on these conditions and proofs). Lemma 1. For a truthful mechanism when v−i is ﬁxed: (a)The price pi (vi , v−i ) cannot depend directly on the declaration vi of player i, but only on his allocation ai (vi , v−i ) and the declarations of the other players, that is pi (vi , v−i ) = pi (ai (vi , v−i ), v−i ). (b) For every player i the outcome ai satisﬁes ai (vi , v−i ) ∈ argmaxai {vi (ai )−pi (ai , v−i )} where the quantiﬁcation is over all the alternatives that i can enforce for diﬀerent vi and ﬁxed v−i . (c) If for ﬁxed v−i the regions where player i has assignment ai and ai , share a common boundary, then any valuation vi on tis boundary satisﬁes vi (ai ) − vi (ai ) = pi (ai , v−i ) − pi (ai , v−i ). A matrix representation of ﬁnite domains. [3] We will denote any ﬁnite domain of valuations D as a set of matrices. We have one matrix for each valuation function v = (v1 , . . . , vn ) : A → R that belongs to the domain. This matrix has one column for each alternative a ∈ A and one row for each player. Thus the valuation vi of player i is a vector (row of the previous matrix) of numbers that has one coordinate for each possible alternative and we denote the set of all possible such vectors for player i by Vi . (The domain is the set of all possible inputs of the mechanism.) Under this notation the domain of unrestricted valuations (for which a complete characteri- zation is given in [12]) contains all possible matrices with real values. We will say that Si is a subdomain of Vi if the set of all possible valuation vectors Si is a subset of Vi . We will say that D = S1 × . . . × Sn is a subdomain of D = V1 × . . . × Vn if D ⊆ D . Aﬃne transformations of domains. If D is the matrix representation of a domain we denote by λD + c the following aﬃne transformation of D: Multiply the valuations of each player i by a positive constant λi and add a matrix of constants c, with one row ci for each player and one column for each possible allocation. For example the following is an aﬃne transformation of 2-player combinatorial auctions: c1 ∅ λ1 v1 ({1}) + c1 λ1 v1 ({2}) + c1 λ1 v1 ({1, 2}) + c1 {1} {2} {1,2} . λ2 v2 ({1, 2}) + c2 2 2 {1,2} λ2 v2 ({2}) + c{2} λ2 v2 ({1}) + c{1} c2 ∅ 2 Our results Derivation of the characterization of a domain from the characterization of one of its sub-domains. Suppose we know which mechanisms are truthful for a given domain, does this tell us which mechanisms are truthful for any super-domain of it? The ﬁrst reaction may be: we can read the proofs and produce (tediously) similar ones. But then the mechanism for the bigger domain has to satisfy truthfulness for a superset of the input space. Are then perhaps the mechanisms for the bigger domain a subset of the mechanisms for the sub-domain? We have to be careful: it is true that if a mechanism is truthful for the bigger domain, then its restriction to the smaller domain is a truthful mechanism for the smaller domain (for which we assumed that we know a characterization). However it then remains to describe the mechanism for the additional inputs we allowed by enlarging the domain. Theorem 1. Let V be a sub-domain of the domain of unrestricted valuations and superdomain of the domain of additive valuations. If the only possible n-player stable mechanisms for V are aﬃne maximizers, then the same holds for every super-domain of V .1 We want to show that there is no other way to extend a mechanism, which is an aﬃne maximizer for the smaller domain V , to the bigger domain other than an aﬃne maximizer for the bigger domain. If we did not require the mechanism to be truthful, then there would be many possibilities to extend the mechanism to a mechanism that would not be an aﬃne maximizer for the whole domain. Note that in Theorem 1 we did not assume decisiveness, this is because Lemma 2 shows that by truthfulness the range of the mechanism for the bigger domain is the same as the range of it’s restriction to the subdomain. Lemma 2. Let Si be the domain of additive valuations, or any super-domain of it, and Si ⊆ Vi . Consider a social choice function f (·, v−i ) : V1 × . . . × Vn → A for ﬁxed v−i , and constrain it to the domain S1 × . . . × Sn . If the range of the restricted function is a set of alternatives A, then the same set of alternatives is also the range of the social choice function f (·, v−i ) when it is constrained to the bigger domain S1 × . . . × Vi × . . . × Sn . Lemma 3. Start with an aﬃne maximizer M deﬁned for the domain of valuations S1 × . . . × Sn and then consider the bigger domain S1 × . . . × Vi × . . . × Sn where Vi is such that Si ⊆ Vi . If we concentrate on stable mechanisms, there is a unique way to extend M to a truthful mechanism for the bigger domain, namely an aﬃne maximizer deﬁned by the same λ, γ as M . Aﬃne transformations of domains. Note that the next theorem holds for any choice of the domain D, and not only for the domain of additive valuations. This theorem implies that if we characterize all possible mechanisms for a domain of valuations D then the same characterization holds for all domains we get by translating D. Theorem 2. There is a bijection between the mechanisms for D and the mechanisms of λD + c. That is the mechanism with the same allocation and payments p = λ · p + c is also truthful for λD + c. This holds for any number of players n. Threshold mechanisms and their payments. The characterization in [4] reveals the class of threshold mechanisms, which are truthful, very simple in their description, and not (necessar- ily) aﬃne maximizers. The immediate question is whether there exist other domains for which threshold mechanisms are truthful. We describe here the truthful threshold mechanisms for the translated domain λD + c. Theorem 3. If D is the domain of additive valuations then a mechanism for the domain λD + c is a threshold mechanism if and only if it satisﬁes pi (ai , v−i ) − ci i = m aij pi ({j}, v−i ) − ci . a j=1 {j} How to vanish threshold mechanisms. Here we show how starting from the additive domain and slightly enriching the domain of possible valuations we obtain a domain that does not admit any truthful threshold mechanisms. This shows that truthful threshold mechanisms are speciﬁc for the domain of additive valuations and its aﬃne transformations and that they cannot be generalized for richer domains. 1 The proof of Theorem 1 for the 2-player case, goes along exactly the same lines as the proof of Lemma 3.1 [6] by Dobzinski. (The statement of that Lemma involves a diﬀerent setting, with which we don’t deal with in this paper, that of two-player multi-unit auction.) Let Si be the set of all valuation functions vi that are additive. We deﬁne the set of valuation functions Si +δ as follows: Si +δ contains all valuation functions vi with vi (ai ) = m aij vi ({j})+ j=1 (|Ai | − 1)δ where δ = 0 is some constant. That is vi ∈ Si and vi ∈ Si + δ agree only on the valuation for getting singletons and the emptyset and diﬀer by some multiple of δ for bigger bundles. Only the sign of δ matters so we can set it to a tiny constant. There exist many other choices of valuations for which our proofs hold. However if you would like in the end to get the characterization of auctions whose valuations satisfy a certain property, say sub-modularity, you should of course mind to make a choice of valuations that are submodular. We start with two domains, that diﬀer slightly in the valuations one of the players. Each one separately admits truthful threshold mechanisms, but their union does not: Lemma 4. Consider a truthful mechanism for the domain S1 ∪ (S1 + δ) × S2 × . . . × S2 . If it is a threshold mechanism when restricted to S1 × S2 × . . . × Sn , then it is non-threshold when restricted to (S1 + δ) × S2 × . . . × S2 . Consequently for the domain S1 ∪ (S1 + δ) × S2 × . . . × S2 threshold mechanisms are non- truthful. Theorem 4. If the only truthful mechanisms for the domain S1 × S2 × . . . × S2 are either aﬃne maximizers or threshold mechanisms, then the only truthful stable mechanisms, for the domain S1 ∪ (S1 + δ) × S2 × . . . × S2 , or any super-domain of it, are aﬃne maximizers. Applying our tools for the known characterization. The machinery we just developed opts for a characterization of stable truthful mechanisms for additive combinatorial auctions for n players. We only have one [4] for 2-player mechanisms, that are decisive and allocate all items. The characterization in [4] is only for additive valuations, applying Theorem 2 it also applies to any aﬃne transformation of the domain of additive valuations. (See the Appendix for the statement of the characterization from [4].) We can now state our main Theorem: Theorem 5. The only possible decisive truthful mechanisms for S1 ∪ (S1 + δ) × S2 or any super- domain of it are the aﬃne maximizers. 2-player combinatorial auctions that satisfy free disposal, submodularity, subadditivity (or superadditivity) as well as the 2-player unrestricted domain and 2-player combinatorial public projects are some of the super-domains of S1 ∪ (S1 + δ) × S2 . 3 Conclusion and future directions We used as a black box the characterization of 2-player additive combinatorial auctions [4]. This domain is a sub-domain of all domains we mentioned in this work and our results imply that obtaining this characterization is at least as hard as the characterization of all other domains. Observe that by using in Theorem 1 the characterization for n-player subadditive combinatorial auctions in terms of aﬃne maximizers [11] (which assumes stability and scalability but not deci- siveness) we get that for all super-domains of that domain the only possible mechanisms are the aﬃne maximizers (or similarly using [16] we get a characterization of all superdomains of n-player combinatorial auctions that assumes decisiveness and stability). However these domains do not have 2-player submodular combinatorial auctions as a super-domain. Submodular combinatorial auctions is an important domain [7, 10, 19] whose characterization (assuming decisiveness and that all items are allocated) we obtain in this work for the ﬁrst time almost for free. Allthough we characterize at once the very rich class of super-domains of additive combina- torial auctions, the most important aspect of our work is not in characterizing new domains, but in classifying them and obtaining a uniﬁed understanding. A more important reason why we used this speciﬁc characterization is that it is the only one that involves truthful mechanisms that are not aﬃne maximizers. We enrich the domain very slightly and these mechanisms seize to be truthful, thus the domain of additive combinatorial auctions is the transition domain [19, 15] where the aﬃne maximizers are not any more the only truthful mechanisms. In this way we obtained a classiﬁcation of many important domains in terms of which domain’s characterization we can use as a black box in order to obtain the characterization of all of it’s super-domains. Of course the big open question still remains to obtain characterizations of domains that admit non-stable mechanisms. However the approach of classifying domains in a way similar with the one we propose here provides a more thorough understanding of the existing techniques and results and adds rigor to an intuition that was on the same time helpful and misleading. Can we conjecture that a similar classiﬁcation holds for the general n-player case? Acknowledgements: I would like to thank Giorgos Christodoulou, Elias Koutsoupias and a a Annam´ria Kov´cs for helpful discussions and comments. References 1. Sushil Bikhchandani, Shurojit Chatterji, Ron Lavi, Ahuva Mu’alem, Noam Nisan, and Arunava Sen. Weak monotonicity characterizes deterministic dominant-strategy implementation. Econometrica, 74(4):1109–1132, 2006. 2. Dave Buchfuhrer, Shaddin Dughmi, Hu Fu, Robert Kleinberg, Elchanan Mossel, Christos Papadimitriou, Michael Schapira, Yaron Singer, and Chris Umans. Inapproximability for vcg-based combinatorial auctions. In SODA, 2010. 3. George Christodoulou and Elias Koutsoupias. Mechanism design for scheduling. Bulletin of the European Association for Theoretical Computer Science (BEATCS), 97:39–59, February 2009. 4. George Christodoulou, Elias Koutsoupias, and Angelina Vidali. A characterization of 2-player mechanisms for scheduling. In Algorithms - ESA, pages 297–307, 2008. 5. George Christodoulou, Elias Koutsoupias, and Angelina Vidali. A lower bound for scheduling mechanisms. Algorithmica, 2008. 6. Shahar Dobzinski. A note on the power of truthful approximation mechanisms. CoRR, abs/0907.5219, 2009. 7. Shahar Dobzinski. An impossibility result for truthful combinatorial auctions with submodular valuations. In STOC, pages 139–148, 2011. 8. Shahar Dobzinski and Noam Nisan. Limitations of vcg-based mechanisms. In STOC, 2007. 9. Shahar Dobzinski and Noam Nisan. A modular approach to roberts’ theorem. In SAGT, pages 14–23, 2009. 10. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Approximation algorithms for combinatorial auctions with complement-free bidders. In STOC, pages 610–618, 2005. 11. Shahar Dobzinski and Mukund Sundararajan. On characterizations of truthful mechanisms for combinatorial auctions and scheduling. In EC, 2008. 12. Roberts Kevin. The characterization of implementable choice rules. Aggregation and Revelation of Preferences, pages 321–348, 1979. 13. Elias Koutsoupias and Angelina Vidali. A lower bound of 1+φ for truthful scheduling mechanisms. In MFCS, pages 454–464, 2007. 14. Piotr Krysta and Carmine Ventre. Combinatorial auctions with veriﬁcation are tractable. In ESA (2), pages 39–50, 2010. 15. Ron Lavi. Searching for the possibility-impossibility border of truthful mechanism design. SIGecom Exch., 2007. 16. Ron Lavi, Ahuva Mu’alem, and Noam Nisan. Towards a characterization of truthful combinatorial auctions. In FOCS, pages 574–583, 2003. 17. Ron Lavi, Ahuva Mualem, and Noam Nisan. Two simpliﬁed proofs for roberts theorem. Social Choice and Welfare, 32(3):407–423, 2009. 18. Noam Nisan and Amir Ronen. Algorithmic mechanism design (extended abstract). In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC), pages 129–140, 1999. 19. Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay Vazirani. Algorithmic Game Theory. Cambridge University Press, 2007. 20. Christos H. Papadimitriou, Michael Schapira, and Yaron Singer. On the hardness of being truthful. In FOCS, pages 250–259, 2008. 21. Angelina Vidali. The geometry of truthfullness. In WINE, 2009. A Additional deﬁnitions and preliminaries. An aﬃne maximizer is a mechanism deﬁned by a non-negative weight λi for each player (at least one of the λi s is non-zero) and a vector of constants γ where the number of coordinates of the vector is |A|. The allocation of an aﬃne maximizer is such that f (v) ∈ argmaxa { i λi vi + γ} and 1 pi (v) = − λi j=i λj · vj + γ + h(v−i ). A mechanism is decisive when (for ﬁxed values of the other players) a player can enforce any outcome (allocation), by declaring very high or very low values. Deﬁnition 1 (Threshold mechanism) A threshold mechanism for the additive combinatorial auctions (/scheduling) domain is one for which there are threshold functions hij such that the mechanism allocates item j to player i if and only if vi ({j}) ≥ hij (v−i ). What distinguishes these mechanisms from general mechanisms is that the thresholds depend only on the values of the other players but not on the other values of the player himself. In threshold mechanisms there is a single threshold for getting or not item j and it is the same regardless if the rest of the items allocated to player i.2 We deﬁne the Combinatorial Auction domain as follows: There is a set of m items for sale and n players/bidders. The alternatives are allocations of the items to bidders.3 The valuations of the players additionally satisfy vi (∅) = 0 (normalization) and vi (A) = vi (Ai ) (no externalities). Each item can be allocated to at most one player. This deﬁnition is practically the auction that is closest to unrestricted valuations. In the literature [16, 2] the term combinatorial auctions is used for auctions where the valuations of the players also satisfy free disposal. We will however use the term combinatorial auction for the setting deﬁned above and then show that our characterization also holds if we impose any of the following additional restrictions to the valuation function. Free Disposal: The valuation should be non-decreasing with the set of allocated items, i.e. for every Ai ⊆ Bi we have that vi (Ai ) ≤ vi (Bi ). Sub-additive valuations. A valuation vi is subadditive if for any two sets Ai and Ai , vi (Ai ) + vi (Ai ) ≥ vi (Ai ∪ Ai ). Superadditive valuations: For any two disjoint sets Ai and Ai , vi (Ai ) + vi (Ai ) ≤ vi (Ai ∩ Ai ). Submodular valuations: for any two sets Ai and Ai , vi (Ai ) + vi (Ai ) ≥ vi (Ai ∪ Ai ) + vi (Ai ∩ Ai ). Submodular valuations are a subset of subadditive valuations. The scheduling domain is essentially the same as the domain of combinatorial auctions with additive valuations. The only diﬀerence is that one is a maximization problem and the other minimization. In auctions the utilities of the players are v − p and in scheduling the utilities of the machines are −v + p. Therefore we can restate the characterization Theorem from [4] as follows: Theorem 6 ( [4]). For the combinatorial auction domain D with additive valuations, or any aﬃne transformation of it λD + c the only decisive (decisive for at least 3 outcomes) truthful mechanisms for two players and two items, that allocate all items are either aﬃne maximizers or threshold mechanisms. For the case of more than two items every decisive truthful mechanism for 2 players partitions the items into groups. Items in a group of size at least two are allocated by an aﬃne maximizer 2 It is not true in general that every set of functions hij deﬁnes a legal mechanism, as they have to be consistent between them. In particular, the threshold functions should be such that every item j is allocated to exactly one player. In other words, exactly one of the constraints vi ({j}) ≥ hij (v−i ), for i = 1, . . . , n, should be satisﬁed. 3 We also use the notation ai for the binary vector where aij is 1 if player i gets item j and 0 if he doesn’t. To go from one notation to the other just consider that aij = 1 if j ∈ Ai and 0 else. and items in singleton groups by threshold mechanisms. The allocation of two diﬀerent groups is not entirely independent: The values of the items in a group allocated by an aﬃne maximizer can appear in the threshold mechanism for a diﬀerent group of items. The aﬃne maximizers cannot be aﬀected by the values of the items in a group allocated by a threshold mechanism. Examples of matrix representations of domains with which we deal – Unrestricted valuations: (each one of the valuations can be any real number) v1 (a) v1 (b) v1 (c) v1 (d) v2 (a) v2 (b) v2 (c) v2 (d) v3 (a) v3 (b) v3 (c) v3 (d) – Combinatorial public projects: the valuations are submodular and vi (∅) = 0. (The valuations are restricted but the outcome is the same for all players just like before.) v1 (∅) = 0 v1 ({1}) v1 ({2}) v1 ({1, 2}) v2 (∅) = 0 v2 ({1}) v2 ({2}) v2 ({1, 2}) v3 (∅) = 0 v3 ({1}) v3 ({2}) v3 ({1, 2}) – Combinatorial auctions: the valuations are submodular or subadditive or superadditive or ad- ditive and each item is allocated to exactly one player. v1 (∅) = 0 v1 ({1}) v1 ({2}) v1 ({1, 2}) v2 ({1, 2}) v2 ({2}) v2 ({1}) v2 (∅) = 0 – Additive/Scheduling Domain: v1 (10) v1 (01) v1 (10) + v1 (01) 0 v2 (01) v2 (10) 0 v2 (10) + v2 (01) 10 01 11 00 Setting a = ,b = ,c = ,d = (the names of the alternatives do not 01 10 00 11 matter) we can see that each one of these domains is a subset of the previous domains. B Missing proofs Proof (of Lemma 1). (a) Suppose towards a contradiction that there exist vi , vi such that ai (vi , v−i ) = ai (vi , v−i ), but pi (vi , v−i ) < pi (vi , v−i ). Then when the true processing times of player i are vi he has incentive to declare falsely that his processing times are vi . His valuation remains the same (as we assumed that ai (vi , v−i ) = ai (vi , v−i )) and his payment decreases. Consequently by declar- ing falsely vi his utility increases vi (ai (vi , v−i ), vi ) − pi (vi , v−i ) > vi (ai (vi , v−i ), vi ) − pi (vi , v−i ). This contradicts the assumption that the mechanism is truthful. (b) Suppose towards a contra- diction that there exists a type v such that for some allocation ai we have vi (ai (vi , v−i ), vi ) − pi (ai (vi , v−i ), v−i ) < vi (ai , vi ) − pi (ai , v−i ). If vi is such that ai (vi , v−i ) = ai then player i would have incentive to falsely declare vi . (c) Straightforward from (a) and (b). Proof (of Lemma 2). Suppose towards a contradiction that there exists some alternative a that is in the range of f |S1 ×...×Vi ×...×Sn (·, v−i ) but not in the range of f |S1 ×...×Si ×...Sn (·, v−i ). (We denote by f |D the restriction of f to take input only from the domain D.) Take a suﬃciently large constant M such that M >> maxai {pi (ai , v−i )} and −M << minai {pi (ai , v−i )} deﬁne the additive valuation ui ∈ Si such that ui ({j}) = M if j ∈ ai and ui ({j}) = M if j ∈ ai (by / additivity the valuation for bigger bundles is uniquely determined by the valuation for singletons). For suﬃciently large M player i strictly prefers a to all outcomes b = a. In the mechanism for the bigger domain S1 × . . . × Vi × . . . Sn player i can enforce outcome a by declaring the valuation from Vi for which the mechanism outputs a. (By Lemma 1(a) his payments do no change.) Proof (of Lemma 3). As the mechanism is truthful for player i and applying Lemma 1(b) its allocation ai is such that ai ∈ argmaxai {vi (ai ) − pi (ai , v−i )} where the payments are of the form 1 pi (ai , v−i ) = − λi j=i λj · vj + γ (by Lemma 1(a) the payment of player i for the bigger domain only depends on his allocation, thus it is equal to the payment of the restriction of the mechanism to the smaller domain, where the mechanism is an aﬃne maximizer). Plugging in 1 the payments of player i we get ai ∈ argmaxai {vi (ai ) + λi j=i λj · vj + γ } and consequently ai ∈ argmaxai { i λi · vi + γ}. This means that the allocation and payments of the ﬁrst player are the same as that of an aﬃne maximizer. If there are only 2 players and the allocation of the ﬁrst player also determines the allocation of the second player, we are done. If there are more than two players since the mechanism is stable, for every ﬁxed v−i , if the allocation of player i is ﬁxed (and vi changes), then the allocation of all other players is ﬁxed too. Since for vi ∈ Si the allocation of the other players was that of the aﬃne maximizer a ∈ argmaxa { i λi · vi + γ} it is the same also for vi ∈ Vi as long as the allocation of player i remains the same and v−i is ﬁxed. This holds for every ﬁxed v−i and so the mechanism is in whole an aﬃne maximizer, i.e. a ∈ argmaxa { i λi · vi (ai ) + γ}. If λi = 0 then player i is non-decisive and his payment is not required to satisfy any condition. Applying Lemma 2 his range is the same as for the smaller domain, which was a singleton for ﬁxed v−i , so the player remains non-decisive. Proof (of Theorem 1). The proof is by repeatedly applying Lemma 3 n times, where n is the number of players. We ﬁrst extend the domain of player 1 from S1 to V1 , then the domain of player 2 from S2 to V2 and so on. That is we consider the following domains in series S1 × . . . × Sn , V1 × S2 × . . . × Sn , V1 × V2 × S3 × . . . × Sn , and so on until we ﬁnally extend the domain of the mechanism to V1 × . . . × Vn . Proof (of Theorem 2). It is easy to see that the allocation part of the mechanism is truthful as it still satisﬁes the Monotonicity Property. Take two valuations vi ∈ Vi and vi = λi vi + c ∈ λi Vi + ci we construct a new mechanism that is truthful for λi Vi + ci as follows: The allocation part of the two mechanisms for inputs vi and vi respectively is the same so the payments should be such that the boundaries of the mechanism remain the same. Consider the boundaries of the ﬁrst mechanism between two diﬀerent allocations a, a . When the valuations are vi ∈ Vi we have (by Lemma 1(c)) vi (ai ) − vi (ai ) = p(ai , v−i ) − p(ai , v−i ). Again by Lemma 1(c) the boundaries of the new mechanism are: λi · (vi (ai ) − vi (ai )) + cai − cai = p (ai , v−i ) − p (ai , v−i ). Combining the latter two equations we get λi (p(ai , v−i ) − p(ai , v−i )) + cai − cai = p (ai , v−i ) − p (ai , v−i ). Setting the payments pi of the new mechanism to be pi (ai , v−i ) = λi · pi (ai , v−i ) + cai then they satisfy the previous equation. Consequently starting from a mechanism for D = V1 × . . . × Vn we got a truthful mechanism for V1 × . . . × λi Vi + ci × . . . × Vn with the same allocation as the initial mechanism. Repeating the same argument n times we can get a truthful mechanism for the translated domain λD + c. Proof (of Theorem 3). It is easy to show that the payments of a threshold mechanism for the translated domain should satisfy the given condition. Suppose for the other direction that the payments of the mechanism satisfy the given condition. We will show that it is a threshold mechanism. Fix a player i and the values v−i of the other players, for simplicity we will write p(a) instead of pi (ai , v−i ). Take two allocations a and a , that diﬀer only on task k (i.e. ak = 1 − ak and aj = aj for j = k), then m [p(a ) − ca ] − [(p(a) − ca )] = aj · p({j}) − c{j} + (1 − ak )(p({k}) − c{k} ) j=1 j=k m − aj p({j}) − c{j} + ak (p({k}) − c{k} ) j=1 j=k = (1 − 2ak )(p({k}) − c{k} ) = (−1)ak (p({k}) − c{k} ) Since the mechanism is truthful, if a is the allocation produced by the mechanism when the reported valuations are v, it should be λi v(a) + ca − p(a) ≥ λi v(a ) + ca − p(a ). Taking two allocations a and a , that diﬀer only on task k, rearranging the previous inequality and taking into account that the valuations v ∈ D are additive, we get (−1)ak (λi v({k})+c{k} ) ≥ (−1)ak p({k}) for every k = 1, . . . , m. Let Fa := {v(a) | (−1)a1 (λi v({1}) + ca ) ≥ (−1)a1 pi ({1}), . . . , (−1)am (λi v({m}) + c{m} ) ≥ (−1)am pi ({m})} Let Ra be the subregion of the input space where the mechanism gives assignment a. These sets satisfy Ra ⊆ Fa and Fa ∩ Fb = ∅, for any two allocations a, b with a = b. Finally as the mechanism is a partition of the input space, we get that Ra = Fa . It is now easy to see that the mechanism is a threshold mechanism: player i gets task k if and only if λi vi ({k}) + c{k} ≥ pi ({k}, t−i ). Proof (of Lemma 4). If a mechanism is truthful for S1 ∪ (S1 + δ) × S2 × . . . × Sn , then by Lemma 1(a) the payments of player 1 are the same regardless of whether his valuation is from S1 or from S1 + δ. From Theorem 3 if a mechanism is a threshold mechanism (for a subset of items) for the domain of valuations S1 it satisﬁes p1 (a1 , v−1 ) = m a1j p1 ({j}, v−1 ). Supposing towards j=1 a contradiction that it is a threshold mechanism (for the same or a smaller subset of items) also for S1 + δ, it would, again by Theorem 3, satisfy p1 (a1 , v−1 ) − δ(|A1 | − 1) = m a1j p1 ({j}, v−1 ). j=1 Combining the latter two relations, and since δ = 0, we get that the mechanisms cannot be a threshold mechanism for any subset of two or more items when it is restricted to S1 ×S2 ×. . .×Sn . Proof (of Theorem 4). By Theorem 2 since we assumed that the only truthful mechanisms for S1 × S2 × . . . × S2 are either aﬃne maximizers or threshold mechanisms the same characterization also holds for (S1 + δ) × S2 × . . . × S2 . By Lemma 4 a threshold mechanism for S1 × S2 × . . . × Sn is non-threshold for (S1 + δ) × S2 × . . . × Sn , so by our assumption it can be nothing else than an aﬃne maximizer for (S1 + δ) × S2 × . . . × Sn . Applying Lemma 3 the mechanisms is also an aﬃne maximizer when the domain of player 1 is enlarged, so the only possible stable mechanisms for S1 ∪ (S1 + δ) × S2 × . . . × Sn are the aﬃne maximizers. By Theorem 1 the same characterization also holds for any super-domain. Proof (of Theorem 5). Combining Theorems 6 and 4 we get that the only truthful mechanisms for the domain S1 ∪ (S1 + δ) × S2 are aﬃne maximizers. By Theorem 1 the characterization also holds for any super-domain of it. For appropriate choice of the sign of δ we have that the domain S1 ∪ (S1 + δ) × S2 is a sub-domain of superadditive, subadditive or submodular combinatorial auctions as well as of public projects or the unrestricted domain. By Theorem 4(b) and and Theorem 6 the only decisive mechanisms for S1 ∪ (S1 + δ) × S2 are the aﬃne maximizers and by Theorem 1 the same holds for any super-domain of S1 ∪ (S1 + δ) × S2 . Obviously additive valuation functions satisfy all these conditions. Valuation functions from S1 + δ satisfy for δ > 0 superadditivity and free disposal and for δ < 0 subadditivity and sub- modularity. C A note on presenting diﬀerent domains of valuations as restrictions of the combinatorial auction domain. Any domain is a subset of R|A| , where |A| is the number of possible outcomes. Here we will generalize a representation of the domain of valuations that lead to interesting results for the scheduling problem [5, 4, 21]. Say there are only two items, then the possible outcomes are the four allocations {1, 2}, {1}, {2}, ∅. The natural way to represent the domain of valuations of a player i is to have one axis for the valuation of the player for each one of the outcomes (except for vi (∅), which, assuming that the domain is normalized, only takes one possible value vi (∅) = 0). So on the x-axis we have the valuation vi ({1}) of player i for getting only the ﬁrst item on the y-axis we have the valuation vi ({2}) of the same player for getting only the second item and ﬁnally on the z-axis his valuation vi ({1, 2}) for getting both items. Basically all auction domains we are interested in are diﬀerent restrictions of this domain. Restrictions in which we are usually interested in, is to assume additivity of the valuations (our domain is only the plane z = x + y), or sub-additivity (our domain is the subset of R3 satisfying z ≥ x + y), or free disposal (i.e. z ≥ x and z ≥ y). For example the scheduling domain assumes additive valuations. Finally multi-unit (here two-unit) combinatorial auctions are the subset of R3 where x = y and z ≥ x, this is basically the only important domain whose characterization we won’t get from scheduling for the simple reason that the additive combinatorial auction domain is not its subdomain.